If, let's say a router goes down, I know an alert can generate a WHD ticket. Let's say the router comes back up ~10 minutes later. Will SolarWinds' alerting system send an update to the original ticket and lower the priority, close the ticket, or some other action?

I guess it depends on what "Solarwinds alerting system" you are referring to (Orion?) The only way this could happen is if your alerting system knew the original WHD ticket number and you had some way to get it in the Subject. I'm not sure if you can change the priority but you could change the ticket status.